Amazing, children, aren’t they? Zipping with potential, thirsty for new experiences, so quick to learn. So where do we take our 10-year-old daughter for holidays every summer? To the beach, of course — a narrow strip of sand beside a wide strip of water. Usually, we try to make sure there is a pool, in case Daisy gets bored with swimming in the sea.

This year, we decided to try something more stimulating: a city break in Europe. And the more we thought about it, the more sense it made. Children don’t go on holiday feeling frazzled and ready to flop, like we do; they go feeling supercharged with anticipation and ready to see, touch and taste things they have never encountered before. In cities such as Paris and Barcelona, the world’s greatest art, architecture, music and food is ready to answer their call.

We have road-tested three cities that offer loads for families, visiting not just child-friendly museums and galleries, but dolphin shows and dodgem rides. We’ve identified painless ways to get around, and hotels and restaurants where youngsters will fit in just fine. What’s more, there are collateral benefits for mum and dad. We found that travelling with children brings fresh excitement to familiar places. Climbing the Eiffel Tower or clapping eyes on the Sagrada Familia becomes 10 times more thrilling when you do it to the sound of whispered wows from the ones you love.

STOCKHOLM

YOU CAN get all scientific about Stockholm’s appeal for families. You can talk about the 240 days of paid paternity leave, the pram-friendly buses and the fact that only one third of the city is really city at all: parks and waterways occupy the rest. Or you can be more intuitive about it, the way children themselves tend to be. What Daisy immediately liked was the sea-spangled waterfront, lined with onion-domed mansions in crayon-box colours. “It’s a bit like Balamory,” she said.

The city’s 14 islands, and the surging channels between, offer more than just atmosphere. Water babies can swim, canoe or fish within splashing range of Sweden’s parliament house. And Stockholm is so spacious, it can afford to devote an entire island to kids’ stuff:

Djurgarden, a parkland oasis with an open-air museum, a zoo, an aquarium and roller-coasters that spin you out over the water.

The other essential island is Gamla Stan, Stockholm’s curly-gabled medieval quarter. Even little legs can walk across it in half an hour, and treasure-hunting in the trinket stores is the closest they will ever get to shopping in Diagon Alley.

Is it family-friendly? Are you kidding? Even the trains from the airport have baby-changes. Public transport is free for under-13s at weekends, and you’ll really struggle to find a restaurant that doesn’t like children. However, the sinewy self-reliance of the Swedes means youngsters are less protected than here: hold hands tight by those unfenced quaysides.

Fun, fun, fun: Stockholm has a mind-boggling museums catalogue — 70 in all, from Abba to Vikings. But planning your weekend is simple enough: board the ferry from Gamla Stan (£2, free for under-7s) and make for Djurgarden.

You can explore the island on bikes (from Djurgardsbron Sjocafe, by the bridge — they also rent out canoes), but the main attractions are all within strolling distance. Start with Skansen (00 46-8 442 8000; £3.60, £1.50 for 6-15s), the world’s oldest open-air museum, which transports you to a sepia-tinted Swe-dish village of rescued 19th-century buildings, staffed by craftspeople in period pinafores.

There is a farmstead with piglets to pet, and a Sami encampment, complete with reindeer; but Daisy liked Skansen’s zoo best of all. It includes the Nordic “big five” (wolf, wolverine, brown bear, lynx and elk), and the 2pm feeding round is a ferment of dashing fur. On a sunny day, though, just being here is a tonic. There’s something properly magical about a hilltop kingdom of wolves and windmills overlooking Stockholm’s impeccably stylish city centre.

Down the road, the Vasamuseet (8 519 54800; £5, 75p for 7-15s) has just one, gobsmacking exhibit: the 17th-century warship Vasa, hauled out of the harbour 333 years after it sank on its maiden voyage. The ship is pure Pirates of the Caribbean, writhing with carvings of mermaids and emperors, and crammed into a twilit hall. Daisy swarmed all over the place, peering into portholes and checking out the models and movies that bring the ship’s tragicomic history to life.

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